Search Results

You are looking at 1 - 4 of 4 items for

  • Author: Éric Sopena x
Clear All Modify Search
Open access

Daouya Laïche, Isma Bouchemakh and Éric Sopena

Abstract

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that its set of vertices V(G) can be partitioned into k disjoint subsets V 1, . . . , Vk, in such a way that every two distinct vertices in Vi are at distance greater than i in G for every i, 1 ≤ ik. For a given integer p ≥ 1, the p-corona of a graph G is the graph obtained from G by adding p degree-one neighbors to every vertex of G. In this paper, we determine the packing chromatic number of p-coronae of paths and cycles for every p ≥ 1.

Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of p-coronae of paths and cycles.

Open access

Éric Sopena and Jiaojiao Wu

Abstract

An incidence in a graph G is a pair (v, e) with v ∈ V (G) and e ∈ E(G), such that v and e are incident. Two incidences (v, e) and (w, f) are adjacent if v = w, or e = f, or the edge vw equals e or f. The incidence chromatic number of G is the smallest k for which there exists a mapping from the set of incidences of G to a set of k colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid Tm,n = Cm2Cn equals 5 when m, n ≡ 0(mod 5) and 6 otherwise.

Open access

Olivier Baudon, Julien Bensmail and Éric Sopena

Abstract

The well-known 1-2-3 Conjecture addressed by Karoński, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1, 2, 3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ receive distinct sums of outgoing weights. This result is tight in the sense that some oriented graphs do not admit such an assignment using the weights from {1, 2} only. We finally prove that deciding whether two weights are sufficient for a given oriented graph is an NP-complete problem. These results also hold for product or list versions of this problem.

Open access

Brahim Benmedjdoub, Isma Bouchemakh and Éric Sopena

Abstract

A 2-distance k-coloring of a graph G is a mapping from V (G) to the set of colors {1,. . ., k} such that every two vertices at distance at most 2 receive distinct colors. The 2-distance chromatic number χ 2(G) of G is then the smallest k for which G admits a 2-distance k-coloring. For any finite set of positive integers D = {d 1, . . ., d}, the integer distance graph G = G(D) is the infinite graph defined by V (G) = ℤ and uvE(G) if and only if |vu| ∈ D. We study the 2-distance chromatic number of integer distance graphs for several types of sets D. In each case, we provide exact values or upper bounds on this parameter and characterize those graphs G(D) with χ2(G(D)) = ∆(G(D)) + 1.